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Let $X_1, \ldots, X_n$ i.i.d. from the $\mathrm{Bernoulli}(p)$ distribution. Then $X = X_1 +\ldots+X_n$ follows the $\mathrm{Binomial}(n,p)$ distribution. We wish to estimate $g(p)=p(1-p)$ whose UMVUE is $$\delta=\frac{X(n−X)}{n(n−1)} $$ Find the asymptotic distribution of $\delta$.

I tried:

let $$ \delta^*=\frac{X(n-X))}{n^2}=\frac{X}{n}-\left(\frac{X}{n}\right)^2=\frac{\sum_{i=0}^n X_i}{n}-\left(\frac{\sum_{i=0}^n X_i}{n}\right)^2=\bar X-(\bar X)^2 $$ so, $$ \delta=\frac{X(n-X)}{n(n-1)}=\frac{n}{n-1}\big(\bar X - (\bar X)^2\big) $$ Let, $$h(\mu) = \left(\frac{n}{n-1}\right)\big(\mu-(\mu)^2\big) $$ So, $$h^\prime(\mu)=\frac{n}{n-1}-2\mu\left(\frac{n}{n-1}\right)$$ hence, the asymptotic distribution of $\delta$ is $$ \sqrt n {\big(h(\bar x) - h(\mu)\big)}\xrightarrow{L} N\big(0, (h^\prime(\mu))^2\sigma^2\big)=N\big(0, \big(\frac{n}{n-1}-2\mu(\frac{n}{n-1})\big)^2\big(np(1-p)\big)^2\big) $$ By the Delta method.

But, to use the Delta method, I must prove $\delta$ and $g(p)$ are consistent. How do I prove it? Thank you very much!

anonyx2
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  • The variance of $\delta$ should be pretty straightforward to derive using binomial moments. So why not just check that $\mathsf{Var}\delta\to 0$? – Aaron Hendrickson Mar 21 '21 at 19:43

1 Answers1

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To show that $\delta$ and $g(p)$ are consistent, use the partial fraction decomposition $\delta=\frac{X(n-X)}{n(n-1)}=\frac{X}{n-1}-\frac{X^2}{(n-1)n}$. Taking the limit, we get $\lim_{n\rightarrow\infty}\delta=p-p^2=p(1-p)=g(p)$.

Vons
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